Rigidification of holomorphic germs with non-invertible differential

Abstract

We study holomorphic germs $f:(\mathbb{C}^2, 0) \rightarrow (\mathbb{C}^2,0) with non-invertible differential $df_0$. In order to do this, we search for a modification $\pi:X \rightarrow (\mathbb{C}^2,0)$ (i.e., a composition of point blow-ups over the origin), and an infinitely near point $p \in \pi^{-1}(0)$, such that the germ $f$ lifts to a holomorphic germ $\hat{f}:(X,p) \rightarrow (X,p)$ which is rigid (i.e., the generalized critical set of $\hat{f}$ is totally invariant and has normal crossings at $p$). We extend a previous result for superattracting germs to the general case, and deal with the uniqueness of this process in the semi-superattracting case ($\operatorname{Spec}(df_0)=\{0, \lambda\}$ with $\lambda \neq 0$). We specify holomorphic normal forms for the nilpotent case and for the type $(0,\mathbb{D})$, that is $\operatorname{Spec}(df_0)=\{0, \lambda\}$ with $\lambda$ in the unitary disk $\mathbb{D} \subset \mathbb{C}$, and formal normal forms for the type $(0, \mathbb{C} \setminus \mathbb{D})$.